A Mercerian Theorem for Slowly Varying Sequences

N. Tanović-Miller

Publications de l'Institut Mathématique, _N_S_28 (1980) no. 42, p. 203 .

Résumé

The purpose of this note is to investigate a Mercerian problem for triangular matrix transformations of slowly varying sequences. A statement of this type for the nonnegative arithmetical means $M_p$, was recently proved by S. Aljančić [1], using the evaluation of the inverse of the associated Mercerian transformation. In this note a corresponding result is proved for nonnegative triangular matrix transformations satisfying a certain condition, which can be applied to the arithmetical means $M_p$ $p_n \geq 0$, the Ces\`aro transformation $C_\alpha$ of order $\alpha$, $0\alpha\leq1$, other Nörlund transformations $N_p$, $p_n>0$ and $(p_{n+1}/p_n)$ nondecreasing, as well as to some other standard methods. The proof is based on the properties rather than on the evaluation, of the inverse of the associated Mercerian transformation.